Questions tagged [parabolic-pde]
Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
257
questions with no upvoted or accepted answers
13
votes
0
answers
365
views
Pseudolocality outside of geometric PDE?
In Ricci flow, the pseudolocality theorem says roughly that regularity in some region implies that as time goes on, there is some regularity in a smaller region. The first version is due to Perelman. ...
- 856
8
votes
0
answers
441
views
Regularity result for the boundary value problem for the heat equation
Let $\Omega$ be an open bounded subset of $\mathbb R^N$.
Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$
Consider the following boundary value problem for the heat equation:
...
user139845
6
votes
0
answers
149
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
- 61
6
votes
0
answers
109
views
Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
6
votes
0
answers
203
views
Interior regularity for parabolic systems in divergence form
Let $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, be a smooth, bounded domain. Suppose $N \in \mathbb N$, $D \subset \mathbb R^N$ and that $a_{ij} : D \to \mathbb R$ are smooth for $i, j \in \{1, \...
- 303
6
votes
0
answers
179
views
Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary
I'm looking for a reference for a statement like:
Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form
$\mathcal L ...
6
votes
0
answers
222
views
Curvature decay of Ricci expanders
Let $M$ be a gradient Ricci expander with nonnegative curvature operator. Assume $\Sigma$ is its space of directions at infinity (so $M$ looks like a cone over $\Sigma$).
What is the curvature ...
- 43.7k
6
votes
0
answers
372
views
Linear PDE with non constant coefficients and properties of Green's Function
Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...
- 61
6
votes
1
answer
1k
views
Regularity of solution to Fokker Planck equation
Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t =...
- 339
5
votes
0
answers
160
views
Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)
Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$
$$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$
representing the Fokker-Planck evolution equation for the ...
- 91
5
votes
0
answers
111
views
$L^p$ estimates for linear parabolic pdes
Let $u$ solve the linear parabolic equation
$$
u_t - \Delta u = f \text{ on } \Omega \times (0,T)
$$
with initial condition $u(0)=u_0$ and homogeneous Dirichlet boundary condition on $\partial \Omega ...
- 248
5
votes
0
answers
195
views
Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs
Question:
Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$:
$$
\frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t),
$$
with smooth initial data $...
- 301
4
votes
0
answers
120
views
Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
user516086
4
votes
0
answers
106
views
Continuity of solutions of Elliptic PDE with respect to parameters
Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...
4
votes
0
answers
100
views
Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
- 2,710
4
votes
0
answers
102
views
The logistic elliptic equation
Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
- 1,330
4
votes
0
answers
143
views
Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
- 1,230
4
votes
0
answers
304
views
Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
- 213
4
votes
0
answers
102
views
$\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains
I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary,
$$
\left\{
\begin{...
- 173
4
votes
1
answer
452
views
A variant to the Fokker–Planck equation
Consider the PDE of $p(t,x)\ge 0$ given as
$$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$
with initial and boundary conditions $p(0,\cdot)=\rho$...
- 1,230
4
votes
0
answers
175
views
$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
- 287
4
votes
0
answers
100
views
Global existence of $L^p$-solutions to a quasilinear diffusion equation
We consider the diffusion problem
$$\begin{cases}
\partial_t u = \nabla \cdot (a(u)\nabla u), \quad t>0, x \in \mathbb{R}^n \\
u(0) = u_0
\end{cases}$$
for functions $u \colon [0,T] \times \mathbb{...
- 101
4
votes
0
answers
110
views
Is there a better reference for existence/regularity for parabolic PDEs (and systems) than the book of Ladyzenskaja, Solonnikov, Uralceva?
The book of Ladyzenskaja, Solonnikov, Uralceva contains almost everything most people need yet the typesetting and notation is disgusting to the eye. Is there any better text that covers the same type ...
4
votes
0
answers
184
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
- 5,163
4
votes
0
answers
107
views
Biharmonic heat flow on compact manifolds
Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...
- 3,554
4
votes
0
answers
613
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
- 1,330
4
votes
0
answers
206
views
How to prove that every weak solution is classic for a simple parabolic equation
Consider equation
$$
Lu=u_t-a(x)u_{xx}=0, \tag 1
$$
where coefficient $a$ is bounded and $a(x)\ge \delta>0$ for all $x\in \mathbb R$.
If $a$ is not smooth the conjugate operator $L^*$ can not be ...
- 2,645
4
votes
0
answers
126
views
Approximation argument in geometric flows
I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...
- 435
4
votes
0
answers
181
views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
- 411
4
votes
0
answers
94
views
One-dimensional harmonic map flow with low regularity
My question is the following:
What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in ...
- 9,591
4
votes
0
answers
238
views
Space-time Poincaré inequality for solution of parabolic equation
If $u : \mathbb R^n \to \mathbb R$ is a smooth enough function then on any Euclidean $n$-ball $B_R$ of radius $R$ we have the very well-known Poincaré inequality
$$ \int_{B_R} |u - \bar u|^2 \le C(R,...
4
votes
0
answers
96
views
Existence and uniqueness of an elliptic equation coupled with a parabolic equation (mean curvature flow)
Given a parabolic equation on a simply connected smooth domain $\Omega(t)$ with boundary $\Gamma(t)$
$$
\Delta u = 1 \quad on \quad \Omega(t)
\\
\nabla u \cdot n + u = g \quad on \quad \Gamma(t)
$$
(...
- 41
4
votes
0
answers
164
views
Heat equation violating complementing condition
I am interested in parabolic (initial-)boundary value problems violating the so called "complementing condition" (as defined in e.g. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uralceva, MR 241822 ...
- 41
4
votes
0
answers
495
views
Properties of the solution of the heat equation
Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...
- 679
4
votes
0
answers
206
views
Reference for short time existence of paraobolic PDE on bundles
I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...
- 141
4
votes
0
answers
146
views
Hopf Lemma for strong solutions
The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and $\partial\...
- 41
3
votes
0
answers
69
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
- 161
3
votes
0
answers
66
views
Are solutions of the forced Navier–Stokes equation less regular than those of the Stokes equation?
Let $\Omega \subset \mathbb R^3$ be a smooth, bounded domain and $T > 0$. Let us consider
\begin{align*}
\begin{cases}
u_t + \kappa (u \cdot \nabla) u = \Delta u + \nabla P + f(x, t), \quad \...
- 303
3
votes
0
answers
116
views
Wellposedness of this parabolic PDE
Consider a terminal-boundary value problem for $v: (t,x,y)\in [0,T]\times \mathbb R^2_+\to \mathbb R\ni v(t,x,y)$:
$$
\begin{cases}
v_t + \max(v_x,v_y)+ \frac 1 2 (v_{xx}+v_{yy})=0, & \forall (t,...
- 975
3
votes
0
answers
93
views
Comparison principle for Elliptic PDE with exponential nonlinearity
Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...
3
votes
0
answers
189
views
A non-linear PDE $v^2v_t=v_{xx}v-v_{x}^2$
PS : Indeed, there is a typo in my equation. Thanks to Zachary's observation.
Consider a PDE for $v: [0,1]^2\to (-\infty,0]$ satisfying
$$v_t(t,x) = \frac{v_{xx}(t,x)v(t,x)-v_{x}(t,x)^2}{v(t,x)^2},\...
- 975
3
votes
0
answers
101
views
Uniqueness continuation property for parabolic equation
Consider the following parabolic equation:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial \rho }{\partial t}-\Div\left( a\left( x\right) \nabla
\rho \right) +p(x)\rho = 0 & \...
3
votes
0
answers
121
views
Is the normalized Ricci flow real analytic in the time variable?
Let $(M^n,g)$ be a closed Riemannian manifold. In this paper, B. Kotschwar proved that the Ricci flow $g(t)$ with initial condition $g(0) = g$ is real analytic with respect to the time variable, for $...
- 2,085
3
votes
0
answers
118
views
Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
- 225
3
votes
0
answers
44
views
An equality satisfied by the solutions to Kolmogorov forward and backward PDEs
Let $b: \mathbb R_+\times\mathbb R\to \mathbb R$ and $\sigma: \mathbb R_+\times\mathbb R\to (0,\infty)$ be functions as nice as possible (e.g. bounded and of bounded partial derivatives, and $\inf_{(t,...
- 1,230
3
votes
0
answers
101
views
On the derivatives of the solutions of the heat equations with Neumann boundary condition
Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
- 701
3
votes
0
answers
49
views
Conditions of parameters to have bounded solution of Dynkin's equation in exit problem
Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
- 91
3
votes
0
answers
159
views
Does the weak formulation of a parabolic PDE applies to a (good) non-test function?
Let $\rho:\mathbb R^d\times[0,\infty)\to(0,\infty)$ such that $\int \rho_t(x)\,dx=1$ for all $t\geq0\,$, $\rho$ is Holder-continuous (in both variables) and $\rho_t\in W^{1,1}(\mathbb R^d)$ for a.e. $...
- 311
3
votes
0
answers
127
views
How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?
I am reading a textbook on PDE (Introduction to Elliptic and Parabolic Partial Differential Equations by Z. Wu, J. Yin and C. Wang, written in Chinese) where a proof of the existence of weak solutions ...
- 51
3
votes
0
answers
41
views
Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions
on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem
$$
\partial_t u = \Delta u + ...
- 51